Question

give the center and radius of the circle described by the equation and graph the equation. use the graph to identify the domain and range.
(x + 1)^2+(y - 2)^2 = 9

Explanation:

Step1: Recall circle - standard form

The standard form of a circle equation is $(x - h)^2+(y - k)^2=r^2$, where $(h,k)$ is the center and $r$ is the radius.

Step2: Identify the center

For the equation $(x + 1)^2+(y - 2)^2=9$, we can rewrite it as $(x-(-1))^2+(y - 2)^2=3^2$. So the center $(h,k)=(-1,2)$.

Step3: Identify the radius

Since $r^2 = 9$, then $r=\sqrt{9}=3$.

Step4: Find the domain

The left - most $x$ value of the circle is $x=h - r=-1-3=-4$, and the right - most $x$ value is $x=h + r=-1 + 3=2$. So the domain is $[-4,2]$.

Step5: Find the range

The bottom - most $y$ value of the circle is $y=k - r=2-3=-1$, and the top - most $y$ value is $y=k + r=2 + 3=5$. So the range is $[-1,5]$.

Answer:

Center: $(-1,2)$; Radius: $3$; Domain: $[-4,2]$; Range: $[-1,5]$